I have read this topic. I remember the test paper said that only the final range of m was required. This process is not required. When t takes 0. And 1. There are two extreme cases. Bring the E-point coordinates of extreme cases into y=x2+mx. You will get two extremes of m. Then it can be considered that the conclusion can also be established if m takes the point between the two. So I wrote it like this.

If you need specific proof. I think we should use high school knowledge.